Commit 76d280b9 authored by Ralf Jung's avatar Ralf Jung

add telescopic versions of the Coq quantifiers

parent d070e44a
......@@ -141,3 +141,45 @@ Notation "'λ..' x .. y , e" :=
(tele_app $ tele_bind (λ x, .. (tele_app $ tele_bind (λ y, e)) .. ))
(at level 200, x binder, y binder, right associativity,
format "'[ ' 'λ..' x .. y ']' , e").
(** Telescopic quantifiers *)
Definition tforall {TT : tele} (Ψ : TT Prop) : Prop :=
tele_fold (λ (T : Type) (b : T Prop), x : T, b x) (λ x, x) (tele_bind Ψ).
Arguments tforall {!_} _ /.
Definition texist {TT : tele} (Ψ : TT Prop) : Prop :=
tele_fold ex (λ x, x) (tele_bind Ψ).
Arguments texist {!_} _ /.
Notation "'∀..' x .. y , P" := (tforall (λ x, .. (tforall (λ y, P)) .. ))
(at level 200, x binder, y binder, right associativity,
format "∀.. x .. y , P") : stdpp_scope.
Notation "'∃..' x .. y , P" := (texist (λ x, .. (texist (λ y, P)) .. ))
(at level 200, x binder, y binder, right associativity,
format "∃.. x .. y , P") : stdpp_scope.
Lemma tforall_forall {TT : tele} (Ψ : TT Prop) :
(tforall Ψ) ( x, Ψ x).
Proof.
symmetry. unfold tforall. induction TT as [|X ft IH].
- simpl. split.
+ done.
+ intros ? p. rewrite (tele_arg_O_inv p). done.
- simpl. split; intros Hx a.
+ rewrite <-IH. done.
+ destruct (tele_arg_S_inv a) as [x [pf ->]].
revert pf. setoid_rewrite IH. done.
Qed.
Lemma texist_exist {TT : tele} (Ψ : TT Prop) :
(texist Ψ) (ex Ψ).
Proof.
symmetry. induction TT as [|X ft IH].
- simpl. split.
+ intros [p Hp]. rewrite (tele_arg_O_inv p) in Hp. done.
+ intros. by exists TargO.
- simpl. split; intros [p Hp]; revert Hp.
+ destruct (tele_arg_S_inv p) as [x [pf ->]]. intros ?.
exists x. rewrite <-(IH x (λ a, Ψ (TargS x a))). eauto.
+ rewrite <-(IH p (λ a, Ψ (TargS p a))).
intros [??]. eauto.
Qed.
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